# Distance from a Point to a Line on a Plane

The distance from a point *M*(*x*_{0}, *y*_{0}) to a straight line *Ax + By + C = *0 in the plane is determined by the formula

Here (*x*_{0}, *y*_{0}) is the coordinate of the point *M* and *A*, *B*, *C* are the coefficients of the equation of the straight line.

This formula calculates the perpendicular distance *d = MN* between the point and the line. The formula can be proved as follows.

## Proof

Let point *N*(*x*_{1}, *y*_{1}) be the foot of the perpendicular drawn from the point *M*(*x*_{0}, *y*_{0}) to the line *Ax + By + C = *0. Find the slope of the line *Ax + By + C = *0:

Therefore, the slope is equal to \(k = -\frac{A}{B}.\)

Since the lines *MN* and *Ax + By + C = *0 are perpendicular, the slope of line *MN* is equal to

Hence the equation of the straight line *MN* has the form

Find now the value of *B _{MN}*. Since point

*M*(

*x*

_{0},

*y*

_{0}) belongs to this line, then its coordinates satisfy the equation:

So

Point *N*(*x*_{1}, *y*_{1}) also belongs to the line *MN*, so its coordinates also satisfy this equation:

From here we get

We denote

Then

The distance between points *M* and *N* is determined by the formula

Therefore

Notice, that

Taking into account that the point *N*(*x*_{1}, *y*_{1}) lies on the line *Ax + By + C = *0, we obtain

Find the value of *q* from this equation:

Substituting the value of \(q\) into the formula for the distance, we get

Using some algebra we get the final expression:

## Solved Problems

### Example 1.

Find the distance from the origin to the straight line \({6x - 8y + 5 = 0}.\)

Solution.

The distance from a point *M*(*x*_{0}, *y*_{0}) to the line *Ax + By + C = *0 is determined by the formula

Substitute the coordinates of the origin *O*(0,0) and the coefficients of the line:

### Example 2.

Find the length of the altitude \(BD\) in a triangle with vertices \({A\left({-3,0}\right)},\) \({B\left({2,5}\right)},\) and \({C\left({3,2}\right)}.\)

Solution.

Write the equation of the line \(AC\) in two-point form:

and convert it into the general form:

Calculate now the distance from point \(B\) to line \(AC,\) that is, the length of the segment \(BD:\)

### Example 3.

Find the distance between two parallel lines \(Ax + By + C_1 = 0\) and \(Ax + By + C_2 = 0.\)

Solution.

Let the point \(M\left({x_0,y_0}\right)\) belong to the line \(Ax + By + C_1 = 0.\) The distance from point M to the straight line \(Ax + By + C_2 = 0\) is given by

Since the point M belongs to the first line, we have

Substituting this into the previous expression, we get

### Example 4.

A line drawn through the origin at the same distance from points \(M\left({3,3}\right)\) and \(N\left({5,1}\right).\) Find the equation of this line.

Solution.

Since the line passes through the origin, its equation is \(Ax + By = 0.\) The distances from points M and N to the line are equal, therefore

Expanding the absolute values, we get two solutions (two straight lines):

Find the equation of the first line:

The equation of the second straight line is written as

The location of these lines is shown in the figure below.